Protractor.



J. GOODPELLOW.

PROTBAGTOR.

APPLICATION IILED r3341, 1910.

1 008,814. Patented Nov. 14, 1911.

WITNESSES." mum/r01? COLUMBIA PLANuflRAPl-l 50., WASHINGTON, D. C.

UNITED STATES PATENT OFFICE.

JOSEPH GOODFELLOW, OF VANCOUVER, BRITISH COLUMBIA, CANADA.

PBQTRACTOR.

Application filed February 11, 1910.

To all whom it may concern:

Be it known that I, JOSEPH GooDrELLow, a subject of the King of GreatBritain, and a resident of Vancouver, in the Province of BritishColumbia and Dominion of Canada, have invented a new and ImprovedProtractor, of which the following is a full, clear, and exactdescription.

This invention relates to a new and improved protractor, of a type inwhich the various angles are designated on an arcuate periphery, wherebythey may be transferred to any suitable work; and which is furtherprovided with a supplementary are or curve for determining variouspoints and aiding in the solution of various geometrical problems.

An object of this invention is to provide a device which will be simplein construction, inexpensive to manufacture, simple in its use, andcomparatively accurate in its results.

A further object of this invention is to provide a protractor withvarious curves and reference points whereby various geometrical problemsmay be solved in a simple and convenient manner.

These and further objects, together with the construction andcombination of parts, will be more fully described hereinafter andparticularly set forth in the claim.

Reference is to be had to the accompanying drawings forming a part ofthis specification, in which similar characters of reference indicatecorresponding parts in all the views, and in which- Figure 1 is a faceView of the protractor; Fig. 2 illustrates the method of obtaining theinner are or curve of the protractor; and Fig. 8 is a diagramillustrating the solution of a geometrical problem by means of thisdevice.

Referring more particularly to the separate parts, 1 indicates the bodyof the protractor, which is in the form of a semi-circular disk of anysuitable material, having its outer periphery divided by suitableindicating marks into degrees of a circle, whose center is indicated at2.

The'body portion 1 of the protractor is provided with an extension 3,the outer edge of which runs parallel to the diameter of the protractorarc, and which is suitably divided into units of measure by suitablereference marks.

The body of the protractor 1 has a cut-out Specification of LettersPatent.

Patented Nov. 14, 1911.

Serial No. 543,264.

portion 4, whose base 5 preferably coincides with the diameter of theprotractor arc which passes through the extremities of the protractorarc. The remaining edge or boundary of this cut-out space of theprotractor is formed by a curve 6, which is ob iained in a peculiarmanner. The method of obtaining this curve is more clearly illus tratedin Fig. 2.

A base line A B is assumed, which approximately equals the diameter ofthe protractor arc, and a distance A C laid 0E thereon, which is justone-quarter of the length of A B. A line C D is then drawn at the pointC perpendicular to the line A B. A number of radiating lines are thendrawn from A to the circumference of a circle A D B drawn through thepoints A D B. with the point 2 as a center. These lines intersect thearc of the circle A D B in the points E, F, G, H, K, L, M; N, O, P, B.These radiating lines A E, A F, etc., also intersect the line C D in thepoints 6, f, g, h, 75, Z, m, n, 0, p, 1", respectively. With the pointsE, F, G, H, K, L, etc., as centers, and with radii equal to thedistances Ac, Af, A9, A72, A70, AZ, etc., arcs are struck intersectingthe lines A E, A F, etc., thereby plotting the curve 6. The curve 6 canbe then drawn through these points, and is utilized with the protractorin obtaining the solutions of various geometrical problems. One of theseproblems is indicated in Fig. 3. Let us assume that We wish to findone-third of any given angle X Y Z. The protractor is applied to theangle X Y Z in such a manner that the point 2 coincides with the point Yon the angle, and also so that the line 2 B coincides with the line Y X.The curve 6 will then intersect the line Y Z at a point V. An arc isthen drawn connecting the points X Z, which corresponds to the outerperiphery of the protractor. If a line is now drawn from a point W,which coincides with the point A on the protractor, through the point V,the projection thereof will intersect the arc X Z at a point T. If thepoints Y and T are then connected, the angle T Y Z will equal one-thirdthe angle X Y Z. In order to demonstrate the truth of this proposition,let X Y Z be any angle. Apply the protractor according to instructionsabove, so that its center will coincide with Y, its diameter will fallalong X W, and X W produced. The outer semicircular rim X T Z W will cutX W at X and W.

The inserted curve J V S W will cut Y Z at V; join WV and V, and produceW V to T; join T Y, then the angle T Y Z equals one-third the angle X YZ.

Proof: Since the curve J V S W is such that any line drawn to thecircumference from the point where the curve, diameter and circumferencemeet is cut so that the rectangle contained by the segment interceptedbetween the said point and the circum ference, and the segmentintercepted between the curve and the circumference is equal to thesquare on the radius, therefore WV T times T V equals Y T Therefore,triangles Y T V, T WV Y are similar; thus angle T Y Z equals angle Y WT. Since Y T equals Y W, then the angle Y WV T equals angle Y T WV;therefore, angle T Y Z equals angle Y WV T equals angle Y T W. Further,inasmuch as the exterior angle of a triangle is equal to the sum of thetwo opposite interior angles of the triangle, then the angle Y V WVequals the angle T Y Z plus the angle Y T W, and the angle X Y Z equalsthe angle Y V WV plus the angle Y W V. Therefore the angle X Y Z equalsthe angle T Y Z, plus the angle Y T WV plus the angle Y WV T equalsthree times the angle T Y Z; therefore the angle T Y Z equals one-thirdthe angle X Y Z. This indicates but one of many solutions, which may beworked out with the aid of this protractor.

While the arc of circle utilized in Fig. 2 for determining the points E.F. G., etc., has been shown separate from the outer periphery of theprotractor, the difference therebetween is intended to be so slight thatit practically coincides therewith. It is to be noted that the curve 6varies in distance from the point A from zero up to threefourths thediameter A B, which is substantially the diameter of the periphery arcof the protractor. It is also to be noted that this same curve 6 variesin distances from the semi-circle A D B or substantially from theperiphery arc of the protractor distances varying from Zero up toonefourth of the diameter A B. It can further be proven by a simplegeometrical proposition that the distance A D is twice the distance C A,therefore the arcs by which the curve 6 is determined have radii varyingfrom one-fourth the diameter of the protractor circle to one-half thediameter of the protractor circle. The curve 6 can further berepresented by a mathematical equation, the derivation of which may beworked out as follows :Referring to Fig. 3, make the line V :/:tperpendicular to the line X WV symbolize X W by 2 A (A representing theradius of a circle); if: W by Y; it V by X, and let K represent the lineW V, and N represent the line V T. Then and A Y K N K KN=2AY and 2AY K NT Substituting the value of K N 2AY X Y in??? Now Y 2AY X 2 K N Z 2 X2Y2 X Y X2 Y2 and thus AY X Y ZAY '2 Z 2 N (K N) new X 71W 4A Y 2AYX 2AYX2 +Y2 the axis of X being a tangent and the axis of Y a diameter of thecircle of which A is a radius.

In testimony whereof I have signed my name to this specification in thepresence of two subscribing witnesses.

JOSEPH GOODFELLOWV.

WVitnesses:

JOHN H. BUCHANAN, ANGUS M. MCIVER.

Copies of this patent may be obtained for five cents each, byaddressingthe Commissioner of Patents, Washington, D. C.

